[论文解读] A Nonlinear Solution to Closed Queueing Networks for Bike Sharing Systems with Markovian Arrival Processes and under an Irreducible Path Graph

Quan‐Lin Li, Rui-Na Fan+1 more|arXiv (Cornell University)|Jul 21, 2017

Transportation Planning and Optimization参考文献 29被引用 4

一句话总结

本文提出了一种针对具有马氏到达过程（MAPs）和不可约路径图的自行车共享系统中封闭队列网络的非线性解法，将自行车建模为虚拟客户，将站点/道路建模为虚拟节点。该方法推导出稳态概率的乘积形式解，从而能够准确计算非泊松、突发性及异构用户到达下的问题站点概率和关键性能指标。

ABSTRACT

As a favorite urban public transport mode, the bike sharing system is a large-scale and complicated system, and there exists a key requirement that a user and a bike should be matched sufficiently in time. Such matched behavior makes analysis of the bike sharing systems more difficult and challenging. This paper considers a more general large-scale bike sharing system from two important views: (a) Bikes move in an irreducible path graph, which is related to geographical structure of the bike sharing system; and (b) Markovian arrival processes (MAPs) are applied to describe the non-Poisson and burst behavior of bike-user (abbreviated as user) arrivals, while the burstiness demonstrates that the user arrivals are time-inhomogeneous and space-heterogeneous in practice. For such a complicated bike sharing system, this paper establishes a multiclass closed queueing network by means of some virtual ideas, for example, bikes are abstracted as virtual customers; stations and roads are regarded as virtual nodes. Thus user arrivals are related to service times at station nodes; and users riding bikes on roads are viewed as service times at road nodes. Further, to deal with this multiclass closed queueing network, we provide a detailed observation practically on physical behavior of the bike sharing system in order to establish the routing matrix, which gives a nonlinear solution to compute the relative arrival rates in terms of the product-form solution to the steady-state probabilities of joint queue lengths at the virtual nodes. Based on this, we can compute the steady-state probability of problematic stations, and also deal with other interesting performance measures of the bike sharing system. We hope that the methodology and results of this paper can be applicable in the study of more general bike sharing systems through multiclass closed queueing networks.

研究动机与目标

• 解决大规模自行车共享系统在非泊松、突发性及空间异构用户到达下的分析挑战。
• 利用具有虚拟客户、节点和服务时间的多类封闭队列网络对自行车共享动态进行建模。
• 开发一种路由矩阵，为复杂系统中相对到达率提供非线性解法。
• 利用产品形式解计算问题站点（满或空）的稳态概率。
• 实现对时间非平稳和空间异构到达模式下自行车共享系统的性能分析。

提出的方法

• 在多类封闭队列网络中，将自行车抽象为虚拟客户，将站点/道路抽象为虚拟节点。
• 使用马氏到达过程（MAPs）对用户到达进行建模，以捕捉突发性和非泊松行为。
• 构建一个路由矩阵，编码虚拟节点间相对到达率之间的非线性关系。
• 推导出所有虚拟节点联合稳态队列长度分布的产品形式解。
• 利用产品形式解计算归一化常数 G(NC) 和单个节点的边缘概率。
• 应用该解法计算性能度量，如问题站点概率和平均队列长度。

实验结果

研究问题

• RQ1 如何为具有非泊松用户到达的大规模自行车共享系统构建封闭队列网络模型？
• RQ2 不可约路径图在表示自行车共享系统地理结构中的作用是什么？
• RQ3 如何为多类封闭队列网络中的相对到达率推导出非线性解法？
• RQ4 在MAP到达下，可以使用产品形式解计算哪些性能度量？
• RQ5 时间非平稳和空间异构的用户到达如何影响自行车共享系统中的站点问题性？

主要发现

• 产品形式解能够精确计算多类封闭队列网络中所有虚拟节点联合队列长度的稳态概率。
• 路由矩阵为相对到达率提供了非线性解法，克服了线性模型在复杂系统中的局限性。
• 问题站点（定义为满或空的站点）的稳态概率可表示为所有系统状态上边缘概率之和。
• 数值结果表明，问题站点的概率随系统参数变化，例如在特定条件下，9个站点系统中第9个站点的概率为0.05734。
• 该方法支持一般到达过程下的分析，包括周期性MAPs和时间非平稳模式，超越了泊松或更新过程的假设。
• 该框架可计算站点和道路上的平均队列长度，为系统拥塞和资源分配提供关键洞察。